proof on difference of two squares and odd integers

Question

How would you prove that every odd integer is a difference of two squares?

I re phrased the problem to make it clearer to me: If $k$ is an odd integer then it ca be expressed in the form $a^2-b^2$ where a and b are integers.

So i start by supposing that k is an odd integer and so $k=2l+1$ for some l in the integers. It seems like you need cases for this but im not really sure how to do it. Could someone help me out please, thanks.

See this picture: https://en.wikipedia.org/wiki/Square_number#Properties. — Michael Hoppe, Dec 04 '18 at 19:35

score 0 · Answer 1 · answered Dec 04 '18 at 19:29

0

Hint:

Compute $(n+1)^2-n^2$.

You should get $2n+1$

answered Dec 04 '18 at 19:29

Mason

3,792

score 0 · Answer 2 · answered Dec 04 '18 at 19:29

0

HINT

$$(k+1)^2-k^2=2k+1$$ $$k^2-(k+1)^2=-(2k+1)$$

answered Dec 04 '18 at 19:29

Rhys Hughes

12,842

score 0 · Answer 3 · answered Dec 04 '18 at 19:29

0

Hint: What is the difference of squares of two consecutive numbers?

answered Dec 04 '18 at 19:29

Vasily Mitch

10,129

score 0 · Accepted Answer · answered Dec 04 '18 at 19:32

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Hint: As you mentioned, you have $k = 2l+1$ for some $l$.

From here, note the difference of squares:

$$a^2-b^2 = (a+b)(a-b)$$

Let $a$ and $b$ be two consecutive integers and try to simplify.

answered Dec 04 '18 at 19:32

KM101

7,176

Thanks, would you have to work from 2l+1 and then add $+l^2-l^2$ then factorise – Carlos Bacca Dec 04 '18 at 19:38
You need to simplify $(b+1)^2-b^2$. Try using the difference of squares and simplify the expression. It will lead straight to the answer you want. – KM101 Dec 04 '18 at 19:39
is that working backwards? – Carlos Bacca Dec 04 '18 at 19:40
actually ignore me – Carlos Bacca Dec 04 '18 at 19:41
Nope, it's the continuation of the proof. – KM101 Dec 04 '18 at 19:41

score 0 · Answer 5 · answered Dec 04 '18 at 19:37

0

$$3=4-1$$

$$2n+1=(n+1)^2-n^2$$

answered Dec 04 '18 at 19:37

hamam_Abdallah

62,951

proof on difference of two squares and odd integers

5 Answers5